Solutions for Higher-Order Three-Point Boundary Value Problems

Volume 2009, Article ID 362983,16pages doi:10.1155/2009/362983

Research Article

Existence and Uniqueness of

Solutions for Higher-Order Three-Point Boundary Value Problems

Minghe Pei

and Sung Kag Chang

1Department of Mathematics, Bei Hua University, JiLin 132013, China

2Department of Mathematics, Yeungnam University, Kyongsan 712-749, South Korea

Correspondence should be addressed to Sung Kag Chang,[email protected] Received 5 February 2009; Accepted 14 July 2009

Recommended by Kanishka Perera

We are concerned with the higher-order nonlinear three-point boundary value problems:xⁿ ft, x, x, . . . , xⁿ⁻¹, n≥3,with the three point boundary conditionsgxa, xa, . . . , xⁿ⁻¹a 0;xⁱb μi, i0,1, . . . , n−3;hxc, xc, . . . , xⁿ⁻¹c 0,wherea < b < c, f:a, c×Rⁿ → R −∞,∞is continuous,g, h : Rⁿ → Rare continuous, andμi ∈ R, i 0,1, . . . , n−3 are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.

Copyrightq2009 M. Pei and S. K. Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Higher-order boundary value problems were discussed in many papers in recent years; for instance, see1–22and references therein. However, most of all the boundary conditions in the above-mentioned references are for two-point boundary conditions2–11,14,17–22, and three-point boundary conditions are rarely seen1,12,13,16,18. Furthermore works for nonlinear three point boundary conditions are quite rare in literatures.

The purpose of this article is to study the existence and uniqueness of solutions for higher order nonlinear three point boundary value problem

xⁿf

t, x, x, . . . , xⁿ⁻¹

, n≥3, 1.1

with nonlinear three point boundary conditions g

xa, xa, . . . , xⁿ⁻¹a 0, xⁱb μi, i0,1, . . . , n−3, h

xc, xc, . . . , xⁿ⁻¹c 0,

1.2

wherea < b < c,f:a, c×Rⁿ → R −∞,∞is a continuous function,g, h:Rⁿ → Rare continuous functions, andμi∈R, i0,1, . . . , n−3 are arbitrary given constants. The tools we mainly used are the method of upper and lower solutions and Leray-Schauder degree theory.

Note that for the cases ofaborbcin the boundary conditions1.2, our theorems hold also true. However, for brevity we exclude such cases in this paper.

2. Preliminary

In this section, we present some definitions and lemmas that are needed to our main results.

Definition 2.1. αt, βt ∈ Cⁿa, care called lower and upper solutions of BVP1.1,1.2, respectively, if

αⁿt≥f

t, αt, αt, . . . , αⁿ⁻¹t

, t∈a, c, g

αa, αa, . . . , αⁿ⁻¹a

≤0, αⁱb≤μ_i, i0,1, . . . , n−3, h

αc, αc, . . . , αⁿ⁻¹c

≤0, βⁿt≤f

t, βt, βt, . . . , βⁿ⁻¹t

, t∈a, c, g

βa, βa, . . . , βⁿ⁻¹a

≥0, βⁱb≥μ_i, i0,1, . . . , n−3, h

βc, βc, . . . , βⁿ⁻¹c

≥0.

2.1

Definition 2.2. Let Ebe a subset ofa, c×Rⁿ. We say thatft, x0, x₁, . . . , x_n−1satisfies the Nagumo condition onEif there exists a continuous functionφ:0,∞ → 0,∞such that

ft, x0, x₁, . . . , x_n−1≤φ|xn−1|, t, x0, x₁, . . . , x_n−1∈E, _∞

0

sds

φs ∞. 2.2

Lemma 2.3see10. Letf : a, c×Rⁿ → Rbe a continuous function satisfying the Nagumo condition on

E

t, x0, x1, . . . , x_n−1∈a, c×Rⁿ:γit≤xi≤Γit, i0,1, . . . , n−2 , 2.3

whereγit,Γit:a, c → Rare continuous functions such that

γ_it≤Γit, i0,1, . . . , n−2, t∈a, c. 2.4 Then there exists a constant r > 0 (depending only on γ_n−2t,Γ_n−2tand φtsuch that every solutionxtof 1.1with

γ_it≤xⁱt≤Γit, i0,1, . . . , n−2, t∈a, c 2.5 satisfiesxⁿ⁻¹_∞≤r.

Lemma 2.4. Letφ:0,∞ → 0,∞be a continuous function. Then boundary value problem xⁿxⁿ⁻²φxⁿ⁻¹, t∈a, c, 2.6 xⁿ⁻²a xⁱb xⁿ⁻²c 0, i0,1, . . . , n−3 2.7

has only the trivial solution.

Proof. Suppose thatx₀tis a nontrivial solution of BVP2.6,2.7. Then there existst₀ ∈ a, csuch thatxⁿ⁻²₀ t0 > 0 orxⁿ⁻²₀ t0 < 0. We may assumexⁿ⁻²₀ t0 > 0. There exists t₁∈a, csuch that

t∈a,cmaxxⁿ⁻²₀ t:xⁿ⁻²₀ t1>0. 2.8

Thenxⁿ⁻¹₀ t1 0,xⁿ₀ t1≤0. From2.6we have

0≥x₀ⁿt1 xⁿ⁻²₀ t1φxⁿ⁻¹₀ t1

>0, 2.9

which is a contradiction. Hence BVP2.6,2.7has only the trivial solution.

3. Main Results

We may now formulate and prove our main results on the existence and uniqueness of solutions fornth-order three point boundary value problem1.1,1.2.

Theorem 3.1. Assume that

ithere exist lower and upper solutionsαt, βtof BVP1.1,1.2, respectively, such that

−1ⁿ⁻ⁱαⁱt≤−1ⁿ⁻ⁱβⁱt, t∈a, b, i0,1, . . . , n−2, αⁱt≤βⁱt, t∈b, c, i0,1, . . . , n−2;

3.1

iift, x0, . . . , x_n−1is continuous ona, c×Rⁿ,−1ⁿ⁻ⁱft, x0, . . . , x_n−1is nonincreasing in x_ii0,1, . . . , n−3onD^b_a, andft, x0, . . . , x_n−1is nonincreasing inx_i i0,1, . . . , n− 3onD_b^cand satisfies the Nagumo condition onD^c_a, where

ϕit min

αⁱt, βⁱt , ψ_it max

αⁱt, βⁱt , i0, . . . , n−2, D_a^b

t, x0, . . . , x_n−1∈a, b×Rⁿ :ϕ_it≤x_i≤ψ_it, i0, . . . , n−2 , D_b^c

t, x0, . . . , x_n−1∈b, c×Rⁿ:ϕ_it≤x_i≤ψ_it, i0, . . . , n−2 , D^c_a

t, x0, . . . , x_n−1∈a, c×Rⁿ:ϕ_it≤x_i≤ψ_it, i0, . . . , n−2 ;

3.2

iiigx0, x₁, . . . , x_n−1is continuous onRⁿ, and−1ⁿ⁻ⁱgx0, x₁, . . . , x_n−1is nonincreasing inxi i0,1, . . . , n−3and nondecreasing inx_n−1on_n−2

i0ϕia, ψia×R;

ivhx0, x₁, . . . , x_n−1is continuous onRⁿ, and nonincreasing inx_i i0,1, . . . , n−3and nondecreasing inx_n−1on_n−2

i0ϕic, ψic×R.

Then BVP1.1,1.2has at least one solutionxt∈Cⁿa, csuch that for eachi0,1, . . . , n−2,

−1ⁿ⁻ⁱαⁱt≤−1ⁿ⁻ⁱxⁱt≤−1ⁿ⁻ⁱβⁱt, t∈a, b,

αⁱt≤xⁱt≤βⁱt, t∈b, c. 3.3 Proof. For eachi0,1, . . . , n−2 define

w_it, x

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ψit, x > ψit, x, ϕit≤x≤ψit, ϕit, x < ϕit,

3.4

whereϕ_it min{αⁱt, βⁱt},ψ_it max{αⁱt, βⁱt}.

Forλ∈0,1, we consider the auxiliary equation xⁿt λf

t, w0t, xt, . . . , wn−2

t, xⁿ⁻²t

, xⁿ⁻¹t

xⁿ⁻²t−λw_n−2

t, xⁿ⁻²t

φxⁿ⁻¹t ,

3.5

whereφis given by the Nagumo condition, with the boundary conditions xⁿ⁻²a λ

w_n−2

a, xⁿ⁻²a

−g

w₀a, xa, . . . , w_n−2

a, xⁿ⁻²a

, xⁿ⁻¹a , xⁱb λμi, i0,1, . . . , n−3,

xⁿ⁻²c λ w_n−2

c, xⁿ⁻²c

−h

w₀c, xc, . . . , w_n−2

c, xⁿ⁻²c

, xⁿ⁻¹c .

3.6

Then we can choose a constantM_n−2>0 such that

−M_n−2< αⁿ⁻²t≤βⁿ⁻²t< M_n−2, t∈a, c, 3.7 f

t, αt, . . . , αⁿ⁻²t,0

−

M_n−2αⁿ⁻²t

φ0<0, t∈a, c, f

t, βt, . . . , βⁿ⁻²t,0

M_n−2−βⁿ⁻²t

φ0>0, t∈a, c, 3.8 αⁿ⁻²a−g

αa, . . . , αⁿ⁻²a,0< M_n−2, βⁿ⁻²a−g

βa, . . . , βⁿ⁻²a,0< M_n−2,

3.9 αⁿ⁻²c−h

αc, . . . , αⁿ⁻²c,0< M_n−2, βⁿ⁻²c−h

βc, . . . , βⁿ⁻²c,0< M_n−2.

3.10

In the following, we will complete the proof in four steps.

Step 1. Show that every solutionxtof BVP3.5,3.6satisfies

xⁿ⁻²t< M_n−2, t∈a, c, 3.11

independently ofλ∈0,1.

Suppose that the estimate|xⁿ⁻²t| < M_n−2 is not true. Then there existst₀ ∈ a, c such thatxⁿ⁻²t0 ≥M_n−2 orxⁿ⁻²t0≤ −Mn−2. We may assumexⁿ⁻²t0 ≥M_n−2 . There existst1∈a, csuch that

t∈a,cmaxxⁿ⁻²t:xⁿ⁻²t1≥M_n−2>0. 3.12

There are three cases to consider.

Case 1t1 ∈a, c. In this case,xⁿ⁻¹t1 0 andxⁿt1≤0. Forλ ∈0,1, by3.8, we get the following contradiction:

0≥xⁿt1 λf

t₁, w₀t1, xt1, . . . , wn−2

t₁, xⁿ⁻²t1

, xⁿ⁻¹t1

xⁿ⁻²t1−λw_n−2

t₁, xⁿ⁻²t1

φxⁿ⁻¹t1 λf

t₁, w₀t1, xt1, . . . , w_n−3

t₁, xⁿ⁻³t1

, βⁿ⁻²t1,0

xⁿ⁻²t1−λβⁿ⁻²t1 φ0

≥λ f

t₁, βt1, . . . , βⁿ⁻²t1,0

M_n−2−βⁿ⁻²t1

φ0

>0,

3.13

and forλ0, we have the following contradiction:

0≥xⁿt1 xⁿ⁻²t1φ0≥M_n−2φ0>0. 3.14

Case 2t1a. In this case,

t∈a,cmaxxⁿ⁻²t:xⁿ⁻²a≥M_n−2>0, 3.15

andxⁿ⁻¹a≤0. Forλ0, by3.6we have the following contradiction:

0< M_n−2≤xⁿ⁻²a 0. 3.16

Forλ∈0,1, by3.9and conditioniiiwe can get the following contradiction:

M_n−2≤xⁿ⁻²a, λ

w_n−2

a, xⁿ⁻²a

−g

w0a, xa, . . . , wn−2

a, xⁿ⁻²a

, xⁿ⁻¹a ,

≤λ

βⁿ⁻²a−g

βa, . . . , βⁿ⁻²a,0

< M_n−2.

3.17

Case 3t1c. In this case,

t∈a,cmaxxⁿ⁻²t:xⁿ⁻²c≥M_n−2>0, 3.18

andxⁿ⁻¹c≥0. Forλ0, by3.6we have the following contradiction:

0< M_n−2≤xⁿ⁻²c 0. 3.19

Forλ∈0,1, by3.10and conditionivwe can get the following contradiction:

M_n−2≤xⁿ⁻²c, λ

w_n−2

c, xⁿ⁻²c

−h

w0c, xc, . . . , wn−2

c, xⁿ⁻²c

, xⁿ⁻¹c

≤λ

βⁿ⁻²c−h

βc, . . . , βⁿ⁻²c,0

< M_n−2.

3.20

By3.6, the estimates

xⁱt< Mi: c−aMi1μi, i0,1, . . . , n−3, t∈a, c 3.21 are obtained by integration.

Step 2. Show that there exists M_n−1 > 0 such that every solution xt of BVP 3.5, 3.6 satisfies

xⁿ⁻¹t< M_n−1, t∈a, c, 3.22

independently ofλ∈0,1.

Let

E{t, x0, . . . , x_n−1∈a, c×Rⁿ:|xi| ≤M_i, i0,1, . . . , n−2}, 3.23

and define the functionFλ:a, c×Rⁿ → Ras follows:

F_λt, x0, . . . , x_n−1 λft, w0t, x0, . . . , w_n−2t, x_n−2, x_n−1

x_n−2−λw_n−2t, xn−2φ|xn−1|. 3.24

In the following, we show that Fλt, x0, . . . , x_n−1 satisfies the Nagumo condition on E, independently ofλ∈0,1. In fact, sincefsatisfies the Nagumo condition onD_a^c, we have

|Fλt, x0, . . . , x_n−1|λft, w0t, x0, . . . , wn−2t, xn−2, xn−1 xn−2−λw_n−2t, xn−2φ|xn−1|

≤12M_n−2φ|x_n−1|:φ_E|x_n−1|.

3.25

Furthermore, we obtain _∞

0

s φEsds

_∞

0

s

12M_n−2φsds ∞. 3.26

Thus,F_λsatisfies the Nagumo condition onE, independently ofλ∈0,1. Let

γit −Mi, Γit Mi, i0,1, . . . , n−2, t∈a, c. 3.27

ByStep 1andLemma 2.3, there existsM_n−1>0 such that|xⁿ⁻¹t|< M_n−1fort∈a, c. Since M_n−2andφ_Edo not depend onλ, the estimate|xⁿ⁻¹t|< M_n−1ona, cis also independent ofλ.

Step 3. Show that forλ1, BVP3.5,3.6has at least one solutionx₁t.

Define the operators as follows:

L:Cⁿa, c⊂Cⁿ⁻¹a, c−→Ca, c×Rⁿ, 3.28

by

Lx

xⁿt, xⁿ⁻²a, xb, . . . , xⁿ⁻³b, xⁿ⁻²c ,

Nλ:Cⁿ⁻¹a, c−→Ca, c×Rⁿ, 3.29

by

Nλx Fλ

t, xt, . . . , xⁿ⁻¹t

, Aλ, λμ0, . . . , λμ_n−3, Cλ

, 3.30

with

A_λ:λ w_n−2

a, xⁿ⁻²a

−g

w₀a, xa, . . . , w_n−2

a, xⁿ⁻²a

, xⁿ⁻¹a Cλ:λ

w_n−2

c, xⁿ⁻²c

−h

w0c, xc, . . . , w_n−2

c, xⁿ⁻²c

, xⁿ⁻¹c

. 3.31

SinceL⁻¹is compact, we have the following compact operator:

Tλ:Cⁿ⁻¹a, c−→Cⁿ⁻¹a, c, 3.32

defined by

Tλx L⁻¹Nλx. 3.33

Consider the setΩ {x∈Cⁿ⁻¹a, c:xⁱ_∞< M_i, i0,1, . . . , n−1}.

By Steps1and2, the degree degI−Tλ,Ω,0is well defined for everyλ∈0,1,and by homotopy invariance, we get

degI−T₀,Ω,0 degI−T₁,Ω,0. 3.34

Since the equationx T₀x has only the trivial solution fromLemma 2.4, by the degree theory we have

degI−T1,Ω,0 degI−T0,Ω,0 ±1. 3.35

Hence, the equationxT₁xhas at least one solution. That is, the boundary value problem

xⁿt f

t, w₀t, xt, . . . , w_n−2

t, xⁿ⁻²t

, xⁿ⁻¹t

xⁿ⁻²t−w_n−2

t, xⁿ⁻²t

φxⁿ⁻¹t ,

3.36

with the boundary conditions xⁿ⁻²a w_n−2

a, xⁿ⁻²a

−g

w0a, xa, . . . , wn−2

a, xⁿ⁻²a

, xⁿ⁻¹a , xⁱb μ_i, i0,1, . . . , n−3,

xⁿ⁻²cw_n−2

c, xⁿ⁻²c

−h

w0c, xc, . . . , wn−2

c, xⁿ⁻²c

, xⁿ⁻¹c ,

3.37

has at least one solutionx1tinΩ.

Step 4. Show thatx₁tis a solution of BVP1.1,1.2.

In fact, the solutionx1tof BVP3.36,3.37will be a solution of BVP1.1,1.2, if it satisfies

ϕ_it≤x₁ⁱt≤ψ_it, i0,1, . . . , n−2, t∈a, c. 3.38

By contradiction, suppose that there existst₀ ∈ a, csuch thatx₁ⁿ⁻²t0 > ψ_n−2t0. There existst1∈a, csuch that

t∈a,cmax

x₁ⁿ⁻²t−ψ_n−2t

:xⁿ⁻²₁ t1−ψ_n−2t1>0. 3.39

Now there are three cases to consider.

Case 1t1∈a, c. In this case, sinceψ_n−2t βⁿ⁻²tona, c, we havex₁ⁿ⁻¹t1 βⁿ⁻¹t1 andxⁿ₁ t1≤βⁿt1. By conditionsiandii, we get the following contradiction:

0≥x₁ⁿt1−βⁿt1

≥f

t₁, w₀t1, x₁t1, . . . , wn−2

t₁, xⁿ⁻²₁ t1

, xⁿ⁻¹₁ t1

xⁿ⁻²₁ t1−w_n−2

t₁, x₁ⁿ⁻²t1

φxⁿ⁻¹₁ t1

−f

t₁, βt1, . . . , βⁿ⁻¹t1

≥f

t₁, βt1, . . . , βⁿ⁻¹t1

xⁿ⁻²₁ t1−βⁿ⁻²t1

φxⁿ⁻¹₁ t1

−f

t₁, βt1,· · ·, βⁿ⁻¹t1

xⁿ⁻²₁ t1−βⁿ⁻²t1

φxⁿ⁻¹₁ t1

>0.

3.40

Case 2t1a. In this case, we have

t∈a,cmax

xⁿ⁻²₁ t−ψ_n−2t

:xⁿ⁻²₁ a−βⁿ⁻²a>0, 3.41

and xⁿ⁻¹₁ a ≤ βⁿ⁻¹a. By 3.37 and conditionsi and iii we can get the following contradiction:

βⁿ⁻²a< xⁿ⁻²₁ a, w_n−2

a, xⁿ⁻²₁ a

−g

w₀a, x1a, . . . , wn−2

a, xⁿ⁻²₁ a

, xⁿ⁻¹₁ a

≤βⁿ⁻²a−g

βa, . . . , βⁿ⁻²a, βⁿ⁻¹a

≤βⁿ⁻²a.

3.42

Case 3t1c. In this case, we have

t∈a,cmax

xⁿ⁻²₁ t−ψ_n−2t

:xⁿ⁻²₁ c−βⁿ⁻²c>0, 3.43

and xⁿ⁻¹₁ c ≥ βⁿ⁻¹c. By 3.37 and conditions i and iv we can get the following contradiction:

βⁿ⁻²c< xⁿ⁻²₁ c w_n−2

c, xⁿ⁻²₁ c

−h

w₀c, x1c, . . . , w_n−2

c, xⁿ⁻²₁ c

, xⁿ⁻¹₁ c

≤βⁿ⁻²c−h

βc, . . . , βⁿ⁻²c, βⁿ⁻¹c

≤βⁿ⁻²c.

3.44

Similarly, we can show thatϕ_n−2t≤xⁿ⁻²₁ tona, c. Hence

αⁿ⁻²t ϕ_n−2t≤xⁿ⁻²₁ t≤ψ_n−2t βⁿ⁻²t, t∈a, c. 3.45

Also, by boundary condition3.37and conditioni, we have

αⁱb xⁱ₁ b βⁱb, in−1−2j, j1,2, . . . , n−1

2

, αⁱb≤xⁱ₁ b≤βⁱb, in−2−2j, j1,2, . . . ,

n−2 2

.

3.46

Therefore by integration we have for eachi0,1, . . . , n−2,

−1ⁿ⁻ⁱαⁱt≤−1ⁿ⁻ⁱxⁱ₁ t≤−1ⁿ⁻ⁱβⁱt, t∈a, b,

αⁱt≤xⁱ₁ t≤βⁱt, t∈b, c, 3.47

that is,

ϕit≤x₁ⁱt≤ψit, i0,1, . . . , n−2, t∈a, c. 3.48

Hencex1tis a solution of BVP1.1,1.2and satisfies3.3.

Now we give a uniqueness theorem by assuming additionally the differentiability for functionsf,gandh, and a kind of estimating condition inTheorem 3.1.

Theorem 3.2. Assume that

ithere exist lower and upper solutionsαt, βtof BVP1.1,1.2, respectively, such that

−1ⁿ⁻ⁱαⁱt≤−1ⁿ⁻ⁱβⁱt, t∈a, b, i0,1, . . . , n−2,

αⁱt≤βⁱt, t∈b, c, i0,1, . . . , n−2; 3.49

iift, x0, . . . , x_n−1 and its first-order partial derivatives in x_i i 0,1, . . . , n −1 are continuous ona, c×Rⁿ,−1ⁿ⁻ⁱ∂f/∂xi ≤ 0i 0,1, . . . , n−3onD_a^b,∂f/∂xi ≤ 0 i0,1, . . . , n−3onD^c_band satisfy the Nagumo condition onD^c_a;

iiigx_n−20, x1, . . . , x_n−1 is continuous on Rⁿ and continuously partially differentiable on

i0ϕia, ψia×R, and

−1ⁿ⁻ⁱ∂g

∂x_i ≤0, i0,1, . . . , n−3,

∂g

∂x_n−1 ≤0, on n−2

i0

ϕia, ψia

×R;

3.50

ivhx_n−20, x1, . . . , x_n−1 is continuous on Rⁿ and continuously partially differentiable on

i0ϕic, ψic×R, and

∂h

∂xi ≤0, i0,1, . . . , n−3,

∂h

∂x_n−1 ≥0, on n−2

i0

ϕic, ψic

×R;

3.51

vthere exists a functionγt∈Cⁿa, csuch thatγⁿ⁻²t>0 ona, c,and

γⁿt<

n−1

i0

∂f

∂x_i ·γⁱt, onD^c_a

n−1

i0

∂g

∂x_i ·γⁱa>0, on n−2

i0

ϕ_ia, ψia

×R,

n−1

i0

∂h

∂xi ·γⁱc>0, on n−2

i0

ϕ_ic, ψic

×R,

γⁱb 0, ifn−i: odd, i0,1, . . . , n−3, γⁱb≥0, ifn−i: even, i0,1, . . . , n−3.

3.52

Then BVP1.1,1.2has a unique solutionxtsatisfying3.3.

Proof. The existence of a solution for BVP 1.1, 1.2 satisfying 3.3 follows from Theorem 3.1.

Now, we prove the uniqueness of solution for BVP1.1,1.2. To do this, we letx₁t andx2tare any two solutions of BVP1.1,1.2satisfying3.3. Letzt x2t−x1t. It is easy to show thatztis a solution of the following boundary value problem

zⁿt ⁿ⁻¹

i0

ditzⁱt, 3.53

n−1

i0

aizⁱa 0,

n−1

i0

cizⁱc 0, 3.54

zⁱb 0, i0,1, . . . , n−3, 3.55

where for eachi0,1, . . . , n−1,

d_it ₁

0

∂

∂xi

f

t, x₁t θzt, x₁t θzt, . . . , xⁿ⁻¹₁ t θzⁿ⁻¹t dθ,

a_{i 1}

0

∂

∂xi

g

x₁a θza, x₁a θza, . . . , xⁿ⁻¹₁ a θzⁿ⁻¹a dθ,

c_{i 1}

0

∂

∂xi

h

x₁c θzc, x₁c θzc, . . . , xⁿ⁻¹₁ c θzⁿ⁻¹c dθ.

3.56

By conditionsii,iii, andiv, we have thatdit∈Ca, c, i0,1, . . . , n−3,and

−1ⁿ⁻ⁱd_it≤0, i0,1, . . . , n−3, t∈a, b, d_it≤0, i0,1, . . . , n−3, t∈b, c,

−1ⁿ⁻ⁱa_i≤0, i0,1, . . . , n−3, a_n−1≤0, ci≤0, i0,1, . . . , n−3, c_n−1≥0.

3.57

Now suppose that there exists t₀ ∈ a, c such that zⁿ⁻²t0/0. Without loss of generality assumezⁿ⁻²t0>0, and let

Ω

M:Mzⁿ⁻²t< γⁿ⁻²t, t∈a, c

. 3.58

It is easy to see that 0 ∈ Ωby condition v, hence Ω/∅. Let M0 supΩ. We have that 0 < M₀ < ∞,M₀zⁿ⁻²t ≤ γⁿ⁻²tona, c, and there exists a pointt₁ ∈a, csuch that M₀zⁿ⁻²t1 γⁿ⁻²t1. Furthermoret₁/a, c. In fact, ift₁ a, thenM₀zⁿ⁻¹a≤ γⁿ⁻¹a.

By conditionvand3.55we can easily show that

−1ⁿ⁻ⁱ

M0zⁱt−γⁱt

≤0, i0,1, . . . , n−3, t∈a, b. 3.59

In particular

−1ⁿ⁻ⁱ

M0zⁱa−γⁱa

≤0, i0,1, . . . , n−3. 3.60

Hence

n−1

i0

M₀a_izⁱa≥ⁿ⁻¹

i0

a_iγⁱa>0, 3.61

which contradicts to 3.54. Thus t₁/a. Similarly we can show that t₁/c. Consequently M₀zⁿ⁻¹t1 γⁿ⁻¹t1.

Now, there are two cases to consider, that is

t₁∈a, b or t₁∈b, c. 3.62

Ift₁∈a, b, then by3.59we have

−1ⁿ⁻ⁱ

M₀zⁱt1−γⁱt1

≤0, i0,1, . . . , n−3. 3.63

Thus, by3.53and conditionvwe have

M₀zⁿt1 n−1

i0

M₀d_it1zⁱt1≥ⁿ⁻¹

i0

d_it1γⁱt1> γⁿt1. 3.64

Consequently, by Taylor’s theorem there existst₂∈t1, csuch that

M₀zⁿ⁻²t> γⁿ⁻²t, ∀t∈t1, t₂, 3.65

which is a contradiction.

A similar contradiction can be obtained ift1 ∈b, c. Hencezⁿ⁻²t ≡0 ona, c. By 3.55, we obtainzt≡0 ona, c. This completes the proof of the theorem.

Next we give two examples to demonstrate the application ofTheorem 3.2.

Example 3.3. Consider the following third-order three point BVP:

x−tx

2t²1 x1

3x³−t⁴sintx, t∈−1,1, 3.66 13x−1

x−1₃

−

x−1 1₃ 0,

x0 0,

−1−x1 2x1 x1₃

x1 1₃ 0.

3.67

Let

ft, x0, x₁, x₂ −tx0 2t²1

x₁1

3x³₁−t⁴sintx₂, gx0, x₁, x₂ 13x₁x³₁−x21³,

hx0, x₁, x₂ −1−x₀2x₁x³₁ x₂1³.

3.68

Choose αt −t, βt t and γt t. It is easy to check that αt −t, and βt t are lower and upper solutions of BVP3.66,3.67respectively, and all the assumptions in Theorem 3.2are satisfied. Therefore byTheorem 3.2BVP3.66,3.67has a unique solution xxtsatisfying

t≤xt≤ −t, t∈−1,0, −t≤xt≤t, t∈0,1,

−1≤xt≤1, t∈−1,1. 3.69

Example 3.4. Consider the following fourth-order three point BVP:

x⁴−t²xx x₃

, t∈−1,1, 3.70

−x−1

x−1₃

13x−1 0,

x0 0, x0 0,

−x1−4x1 x1₂

9x1 0.

3.71

Let

ft, x0, x₁, x₂, x₃ −t²x₀x₂x³₂, gx0, x₁, x₂, x₃ −x0x³₁13x₂, hx0, x1, x2, x3 −x0−4x1x²₁9x2.

3.72

Chooseαt −t², βt t² andγt t². It is easy to check thatαt −t², andβt t² are lower and upper solutions of BVP3.70,3.71, respectively, and all the assumptions in Theorem 3.2are satisfied. Therefore byTheorem 3.2BVP3.70,3.71has a unique solution xxtsatisfying

−2≤xt≤2, t∈−1,1,

2t≤xt≤ −2t, t∈−1,0, −2t≤xt≤2t, t∈0,1,

−t²≤xt≤t², t∈−1,1.

3.73

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